Umemura, Hiroshi; Watanabe, Humihiko Solutions of the third Painlevé equation. I. (English) Zbl 0917.34004 Nagoya Math. J. 151, 1-24 (1998). A rigorous and systematic proof of the irreducibility of the third Painlevé equation is given by applying Umemura’s theory on algebraic differential equations to this equation. The proof consists of two parts: (1) to determine a necessary condition for the parameters of the existence of principal ideals invariant under the Hamiltonian vector field associated to the third Painlevé equation; (2) to determine the principal invariant ideals for a parameter where the principal invariant ideals exist. The method is released from complicated calculation, and has been applied to the proof of the irreducibility of other Painlevé equations. Reviewer: Humihiko Watanabe Cited in 21 Documents MSC: 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 34A34 Nonlinear ordinary differential equations and systems 34A05 Explicit solutions, first integrals of ordinary differential equations Keywords:third Painlevé equation; Umemura’s theory PDF BibTeX XML Cite \textit{H. Umemura} and \textit{H. Watanabe}, Nagoya Math. J. 151, 1--24 (1998; Zbl 0917.34004) Full Text: DOI Digital Library of Mathematical Functions: §32.10(iii) Third Painlevé Equation ‣ §32.10 Special Function Solutions ‣ Properties ‣ Chapter 32 Painlevé Transcendents References: [1] C. R. Acad. Sci. Paris 143 pp 1111– (1906) [2] Nagoya Math. J. 119 pp 1– (1990) · Zbl 0714.12009 [3] Bull. Soc. Math. France 28 pp 201– (1900) [4] Funk. Ekv. 30 pp 305– (1987) [5] Diff. Eq. 18 pp 317– (1982) [6] J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 pp 575– (1986) [7] Dokl. Akad. Nauk. BSSR. 32 pp 395– (1988) [8] Japan. J. Math. 5 pp 1– (1979) [9] Diff. Eq. 20 pp 1191– (1984) [10] Nagoya Math. J. 109 pp 63– (1988) · Zbl 0613.34030 [11] Diff. Eq. 14 pp 1510– (1978) [12] Nagoya math. J. 139 pp 37– (1995) · Zbl 0846.34002 [13] Diff. Eq. 11 pp 285– (1975) [14] Diff. Eq. 3 pp 994– (1967) [15] Diff. Eq. 9 pp 1599– (1973) [16] DOI: 10.1063/1.525260 · Zbl 0504.34022 [17] Acta Math. 33 pp 1– (1909) · JFM 40.0098.04 [18] Groupes et algèbres de Lie, Chapitres 4, 5, et 6 [19] DOI: 10.14492/hokmj/1380892594 · Zbl 0833.34005 [21] Nagoya Math. J. 148 pp 151– (1997) · Zbl 0934.33029 [22] Nagoya Math. J. 144 pp 59– (1996) · Zbl 0878.12002 [23] Nagoya Math. J. 117 pp 125– (1990) · Zbl 0688.34006 [24] Algebraic Geometry and Commutative Algebra in Honor of Masayoshi NAGATA pp 771– (1987) [25] Acta Math. 25 pp 1– (1900) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.